4.8 False alarms, detection threshold and coincident observation

Gravitational-wave event rates in initial interferometers is expected to be rather low: about a few per
year. Therefore, one has to set a high threshold, so that the noise-generated false alarms mimicking an event
are negligible.

For a detector output sampled at 1 kHz and processed through a large number of filters, say 103, one
has instances of noise in a year. If the filtered noise is Gaussian, then the probability of
observing an amplitude in the range of to is

where is the standard deviation. The above probability-distribution function implies that the
probability that the noise amplitude is greater than a given threshold is

Demanding that no more than one noise-generated false alarm occur in a year’s observation means that
. Solving this equation for , one finds that in order that false
alarms are negligible in a year’s observation. Therefore, a source is detectable only if its amplitude is
significantly larger than the effective noise amplitude, i.e., .

The reason for accepting only such high-sigma events is that the event rate of a transient source, i.e., a
source lasting for a few seconds to minutes, such as a binary inspiral, could be as low as a few per year, and
the noise generated false alarms, at low SNRs 3–4, over a period of a year, tend to be quite large.
Setting higher thresholds for detection helps in removing spurious, noise generated events. However, signal
enhancement techniques (cf. Section 5) make it possible to detect a signal of relatively low
amplitude, provided there are a large number of wave cycles and the shape of the wave is known
accurately.

Real detector noise is neither Gaussian nor stationary and therefore the filtered noise cannot be
expected to obey these properties either. One of the most challenging problems is how to remove or veto the
false alarm generated by a non-Gaussian and/or nonstationary background. There has been some effort to
address the issue of non-Gaussianity [125] and nonstationarity [263]; more work is needed in this direction.
However, it is expected that the availability of a network of gravitational wave detectors alleviates the
problem to some extent. This is because a high amplitude gravitational wave event will be coincidentally
observed in several detectors, although not necessarily with the same SNR, while false alarms are, in
general, not coincident, as they are normally produced by independent sources located close to the
detectors.

We have seen that coincident observations help to reduce the false alarm rate significantly. The rate can
be further reduced, and possibly even nullified, by subjecting coincident events to further consistency checks
in a detector network consisting of four or more detectors. As discussed in Section 2, each gravitational
wave event is characterized by five kinematic (or extrinsic) observables: location of the source with respect
to the detector and the two polarizations . Each detector in a network measures a
single number, say the amplitude of the wave. In addition, in a network of detectors, there are
independent time delays in the arrival times of the wave at various detector locations,
giving a total of observables. Thus, the minimum number of detectors needed to
reconstruct the wave and its source is . More than three detectors in a network will have
redundant information that will be consistent with the quantities inferred from any three detectors,
provided the event is a true coincident event and not a chance coincidence, and most likely a
true gravitational wave event. In a detector network consisting of detectors, one can
perform consistency checks. Such consistency checks further reduce the number of false
alarms.

When the shape of a signal is known, matched filtering is the optimal strategy to pull out a signal buried
in Gaussian, stationary noise (see Section 5.1). The presence of high-amplitude transients in the data
can render the background nonstationary and non-Gaussian, therefore matched filtering is not
necessarily an optimal strategy. However, the knowledge of a signal’s shape, especially when it
has a broad bandwidth, can be used beyond matched filtering to construct a veto [31]
to distinguish between triggers caused by a true signal from those caused by high-amplitude
transients or other artifacts. One specific implementation of the veto compares the expected
signal spectrum with the real spectrum to quantify the confidence with which a trigger can
be accepted to be caused by a true gravitational wave signal and has been the most powerful
method for greatly reducing the false alarm rate. We shall discuss the veto in more detail in
Section 5.1.